I'm trying to plot the eigenvalue function of the Laplace equation
$$-u_{xx}-u_{yy}=0,\;(x,y)\in (0,1)^2$$ with $$u(x,y)=0$$ on the boundary of the unit square. I have the nine-point stencil $$A=-\frac{1}{3}\left[\begin{array}{ccc}
1 & 1 & 1\\
1 & -8 & 1\\
1 & 1 & 1
\end{array}\right].$$
The equations for the inner points can be written as $Tu=0$. But when I tried to implement the matrix $T$ in Matlab and get the eigenvectors via the eig-function, it doesn't seem to be the right solution. My code for $T$ is
(n is the number of inner points)
M=-8*eye(n)+diag(ones(n-1,1),1)+diag(ones(n-1,1),-1);
M2=diag(ones(n-1,1),1)+diag(ones(n-1,1),-1);
M3=eye(n)+diag(ones(n-1,1),1)+diag(ones(n-1,1),-1);
T=-1/3*(kron(eye(n),M)+kron(M2,M3));
I computed the smallest eigenvalue and eigenvector for n=63, but it's the reflected of what it should be (or at least exactly the opposite of the picture I have from a solution for the Laplace equation).
So I think somewhere I forgot a -1, but I'm not sure where exactly. I already tried to add one in $T$, but that was not the right thing. Could you please help me?
best regards Aione